Finite Group Databasebeta

← #355×D173 #36355239

Subgroups — #355×D173 #36355239

Order 22144.

260 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#124×D173C211yesmaximal, characteristic
2#101×D173C212yesmaximal, complemented
3#26011288C211yesmaximal, characteristic, complemented
4#26011287C212yesmaximal, complemented
5#355×C173C211yesmaximal, characteristic, complemented
6C2×Q8×D173C2211yescharacteristic
7C2×D4×D173C411yescharacteristic
8M16×D173—22no—
9#19149668C2211yescharacteristic
10#19149667C411yescharacteristic
11#19149666C411yescharacteristic
12#19149665—22no—
13C22⋊C4×D173—22no—
14C42×D173C411yescharacteristic
15#19149598C2212yescomplemented
16#19149597—22no—
17#124×C173C2211yescharacteristic
18#101×C173C2212yescomplemented
19C23×D173—21no—
20C2×Dic346C2×C411yescharacteristic
21C2×Dic346D412yes—
22Q8×D173—41no—
23Q8×D173—41no—
24C2×D692C2×C411yescharacteristic
25C2×C22⋊D173—21no—
26D4×D173—41no—
27D4×D173—41no—
28C2×C4×D173—21no—
29C2×C4×D173C2×C411yescharacteristic
30C2×C4×D173D412yes—
31#14583810—22no—
32#14583809—42no—
33C8×D173—42no—
34C22⋊Dic173—22no—
35#14583803—22no—
36#14583803—22no—
37C4×Dic173—21no—
38C4×Dic173C2×C411yescharacteristic
39C2×Q8×C173C2311yescharacteristic
40C2×D4×C173C2×C411yescharacteristic
41M16×C173—22no—
42C22⋊C4×C173—22no—
43C42×C173C2×C411yescharacteristic
44C22×D173—21no—
45C22×D173—44no—
46C22×D173—41no—
47C22×D173—41no—
48C22×D173C22⋊C411yescharacteristic
49Dic346—41no—
50Dic346—42no—
51D692—41no—
52C22⋊D173—42no—
53C2×Dic173—21no—
54C2×Dic173C22⋊C411yescharacteristic
55C2×Dic173C22⋊C412yes—
56C4×D173—22no—
57C4×D173—21no—
58C4×D173—21no—
59C4×D173—22no—
60C4×D173—42no—
61#13063436—42no—
62C23×C173—21no—
63Q8×C173—41no—
64D4×C173—41no—
65C2×C4×C173—21no—
66C2×C4×C173C22×C411yescharacteristic
67C2×C4×C173C2×D412yes—
68C8×C173—42no—
69C22×C173—42no—
70C22×C173—41no—
71C22×C173C2×C22⋊C411yescharacteristic
72C2×D173—22no—
73C2×D173—42no—
74C2×D173—41no—
75C2×D173—82no—
76C2×D173—82no—
77C2×D173#9911yescharacteristic
78C2×D173#9911yescharacteristic
79C4×C173—22no—
80C4×C173—21no—
81C4×C173—41no—
82Dic173—21no—
83Dic173—22no—
84Dic173—41no—
85C2×C173—21no—
86C2×C173—81no—
87C2×C173C2×#9911yescharacteristic
88D173—21no—
89D173—81no—
90D173#35512yesdirect product factor
91C173C2×#35511yescharacteristic, complemented, Sylow
92C2×#355—1731nomaximal, Sylow
93C2×#124—1731no—
94C2×#101—1732no—
95#355—1731no—
96#355—1732no—
97#355D17311yescharacteristic, direct product factor
98C22×Q8—1731no—
99C22×D4—1731no—
100C2×M16—3462no—
101#124—3462no—
102#124—1731no—
103#124—1731no—
104#124—1731no—
105#124C2×D17311yescharacteristic
106C2×C22⋊C4—3462no—
107C2×C42—1731no—
108#101—3462no—
109#101—1732no—
110#101C2×D17312yescomplemented
111C24—3461no—
112C2×Q8—1731no—
113C2×Q8—1732no—
114C2×Q8—6921no—
115C2×Q8—6921no—
116C2×Q8C22×D17311yescharacteristic
117C2×D4—3461no—
118C2×D4—1731no—
119C2×D4—6921no—
120C2×D4—6921no—
121C2×D4C4×D17311yescharacteristic
122C22×C4—3461no—
123C22×C4—1731no—
124C22×C4—1732no—
125M16—22no—
126M16—3462no—
127M16—6922no—
128C2×C8—6922no—
129C22⋊C4—22no—
130C22⋊C4—3462no—
131C22⋊C4—3462no—
132C22⋊C4—3462no—
133C42—3461no—
134C42—1731no—
135C42C4×D17311yescharacteristic
136C23—21no—
137C23—3461no—
138C23—1731no—
139C23—6924no—
140C23—6921no—
141C23—6921no—
142Q8—41no—
143Q8—6922no—
144Q8—6921no—
145D4—41no—
146D4—6921no—
147D4—6922no—
148C2×C4—21no—
149C2×C4—3462no—
150C2×C4—3462no—
151C2×C4—3461no—
152C2×C4—3461no—
153C2×C4—3461no—
154C2×C4—1731no—
155C2×C4—1732no—
156C2×C4—6922no—
157C2×C4C2×C4×D17311yescharacteristic
158C2×C4D4×D17312yes—
159C8—42no—
160C8—6922no—
161C22—42no—
162C22—41no—
163C22—3462no—
164C22—13842no—
165C22—13842no—
166C22—1731no—
167C22—1731no—
168C22—6922no—
169C22—6921no—
170C22C22⋊C4×D17311yescharacteristic
171C4—22no—
172C4—21no—
173C4—41no—
174C4—3461no—
175C4—3462no—
176C4—6921no—
177C2—21no—
178C2—81no—
179C2—3461no—
180C2—13841no—
181C2—1732no—
182C2#99×D17311yescharacteristic
subgroupG/H
centerC2#99×D173
commutatorC2×C4×C173C22×C4
FrattiniC2×Q8C22×D173
Fitting#355×C173C2
socleC2×C173C2×#99
prime radicalC2×D4×D173C4
cosocleC2×Q8×C173C23
hypercenter#355D173
hypocenterC173C2×#355
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #101×D173C2semidirect
3. #26011288C2semidirect
4. #26011287C2semidirect
5. #355×C173C2semidirect
15. #19149598C22semidirect
18. #101×C173C22semidirect
90. D173#355direct (N × H)
91. C173C2×#355semidirect
97. #355D173direct (N × H)
110. #101C2×D173semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#124×D1732
#101×D1732
#260112882
#260112872
#355×C1732
C2×#355173